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A245055
Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.
0
1, 7, 4, 2, 6, 5, 2, 3, 1, 1, 0, 3, 3, 5, 1, 5, 4, 3, 5, 2, 4, 8, 9, 0, 4, 8, 0, 6, 9, 4, 1, 2, 9, 8, 6, 4, 1, 1, 5, 4, 4, 3, 7, 9, 8, 9, 8, 3, 8, 1, 0, 4, 6, 2, 8, 1, 4, 2, 9, 0, 4, 7, 9, 5, 7, 4, 6, 5, 5, 5, 0, 3, 8, 7, 0, 0, 8, 1, 3, 5, 0, 8, 6, 8, 0, 5, 8, 1, 4, 7, 4, 1, 7, 5, 2, 4, 7, 8, 8, 1, 2
OFFSET
1,2
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 273.
LINKS
Carl Pomerance, The expected number of random elements to generate a finite abelian group, Periodica Mathematica Hungarica 43 (2001), 191-198.
FORMULA
tau = sum_{j >= 1} (1-(1-2^(-j))*prod_{k >= j+1} zeta(k)^(-1)).
tau = sum_{j >= 1} (1-(1-2^(-j))*c*prod_{k = 2..j} zeta(k)), where c is A068982.
EXAMPLE
1.7426523110335154352489048069412986411544379898381...
MATHEMATICA
digits = 101; max = 400; c = 1/Product[N[Zeta[k], digits + 100], {k, 2, max}]; p[j_] := Product[N[Zeta[k], digits + 100], {k, 2, j}]; tau = Sum[1 - (1 - 2^-j)*c*p[j], {j, 1, max}]; RealDigits[tau, 10, digits ] // First
PROG
(PARI) default(realprecision, 120); suminf(j=1, 1-(1-2^(-j))*prodinf(k=j+1, 1/zeta(k))) \\ Vaclav Kotesovec, Oct 22 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved